Let $ p$ be a nonzero polynomial with integral coefficients，Prove that there are at most finitely many numbers $ n$ for which $$ p(n)>2^{n+1}$$
WLOG,I think take $p(x)=a_{0}+a_{1}x+a_{2}x^2+\cdots+a_{k}x^k$,then it is show that there are at most finitely many numbers $ n$ for which $$ a_{0}+a_{1}n+a_{2}n^2+\cdots+a_{k}n^k>2^{n+1}$$ that's why?
Since $\lim_{n \to \infty} \frac{p(n)}{2^{n+1}}=0$, there exists $N \in \mathbb N$ such that $\frac{p(n)}{2^{n+1}}<1$ for all $n\geq N$, or equivalently, $p(n)<2^{n+1}$ for all $n\geq N$.